Abstract

A tachyonic spectral fit to the pulsed γ-ray flux of the Crab is performed, based on the superluminal spectral densities generated by ultra-relativistic electrons. The γ-ray wideband of the pulsar can be reproduced by a tachyonic cascade spectrum, capable of generating the spectral curvature in double-logarithmic plots as well as the extended spectral plateau defined by COMPTEL and EGRET flux points in the MeV band. Estimates on the magnetospheric electron population generating the radiation are inferred from the spectral fit, such as power-law indices, electron temperature and the electronic source count.

1. Introduction

The goal is to point out evidence for the γ-ray spectrum of the Crab pulsar to be tachyonic. The most pronounced difference between electromagnetic and tachyonic radiation lies in the emission process. Due to the negative mass-square of the radiation field, there is a residual radiation that persists in the limit of large orbital bending radius, so that freely moving electrons can radiate superluminal quanta. Radiation densities attached to uniformly moving charges have no analog in electrodynamics, their very existence has substantial implications for the spacetime structure, requiring an absorber medium [1], [2] and [3]. The Green function of the superluminal radiation process is time-symmetric, there is no retarded propagator supported outside the lightcone. The advanced modes of the radiation field are converted into retarded ones by virtue of a nonlocal, instantaneous interaction with the absorber. The latter defines a universal frame of reference, necessary to render the superluminal signal transfer causal. The time symmetry of the Green function implies that there is no radiation damping, the radiating charge stays in uniform motion, the energy radiated is supplied by the oscillators constituting the absorber medium. The superluminal spectral densities depend on the velocity of the charge in the rest frame of the absorber. I also mention that tachyons and photons couple only indirectly via matter fields, there is no absorption of high-energy tachyons by the infrared background. At γ-ray energies, the speed of tachyons is very close to the speed of light, the noticeable difference to electromagnetic radiation is the longitudinally polarized flux component. The polarization of tachyons can be determined from transversal and longitudinal ionization cross-sections, which peak at different scattering angles.

These features of tachyonic wave propagation have been elaborated in previous papers, here we focus on a specific example, a tachyonic spectral fit to the pulsed γ-ray flux of the Crab. When tachyonic spectral densities are averaged with thermal or power-law electron densities, this leads to cascade spectra, extended spectral plateaus followed by power-law decay terminating in exponential decay. A glance at the γ-ray broadband of the Crab pulsar suggests that it can readily be fitted by a cascade spectrum, cf. the figures in Section 3.

In Section 2, we explain the transversal and longitudinal spectral densities of uniformly moving charges, as well as their averaging with thermal and exponentially cut power-law electron distributions. The spectral averaging can be done in closed form, and high- and low-temperature expansions of the tachyonic spectral densities are derived. The spectral breaks and slopes of tachyonic cascade spectra are discussed. In Section 3, we perform the spectral fit to the pulsed Crab γ-wideband, infer the cutoff temperature of the electronic/protonic source populations, and compare it to the break energies in the cosmic-ray spectrum. In Section 4, we present our conclusions.

2. Superluminal spectral densities and spectral averages

The transversal and longitudinal tachyonic radiation densities of a uniformly moving electron read [3]

where Δ^T = 1 and Δ^L = 0. γ is the electronic Lorentz factor, α_q the tachyonic fine structure constant, and m_t the tachyon mass. The spectral cutoff occurs at . Only frequencies in the range 0 < ω < ω_max(γ) can be radiated by a uniformly moving charge, the tachyonic spectral densities p^T,L(ω) are cut off at the break frequency ω_max. The units  = c = 1 can easily be restored. We use the Heaviside–Lorentz system, so that α_q = q²/(4πc) ≈ 1.0 × 10⁻¹³. The tachyon mass is m_t ≈ 2.15 keV/c², the tachyon–electron mass ratio m_t/m ≈ 1/238. These estimates are obtained from hydrogenic Lamb shifts [4]. The tachyonic count rates (tachyons emitted per unit time and unit frequency interval) are n^T,L(ω)  p^T,L(ω)/ω.

The radiation densities (1) refer to a single charge with Lorentz factor γ. We average them with electron distributions, power-laws exponentially cut with Boltzmann densities, dρ ∝ E^−2−αe^−E/(kT)d³p, or

where β  mc²/(kT). At α = −2, the distribution is thermal. The electronic Lorentz factors range in an interval γ₁  γ < ∞. The normalization factors A_α,β(γ₁, n₁) of densities (2) are determined by , where n₁ is the electron count and γ₁ the smallest Lorentz factor of the source population.

The averaging is carried out via

with ω_max(γ) defined above. These averages can be reduced to the spectral functions

where and Δ^T,L as in (1).

The spectral range of the radiation densities (1) is 0 < ω < ω_max(γ). Inversely, the condition or defines the minimal electronic Lorentz factor for radiation at this frequency. ( stands for the rescaled frequency ω/m_t.) That is, an electron in uniform motion can radiate at ω only if its Lorentz factor exceeds . The lower edge of Lorentz factors in the electronic source distribution defines the break frequency, ω₁  ω_max(γ₁), or , which separates the spectrum into a low- and high-frequency band. We have , and if ω > ω₁. The averaged energy density (3) can thus be assembled as

The superscripts T and L denote the transversal and longitudinal polarization components. If the polarization is not distinguished, we add the polarization components, writing p^T+L _α, β for the unpolarized density.

To obtain the low-temperature expansion, β  1, of the spectral average (5), we substitute the asymptotic expansion of the incomplete Γ-function [5] into the spectral function B^T,L(ω;γ;α,β),

The expansion parameter is βγ  1. The high-temperature expansion, β  1, of B^T,L in (4) is based on the ascending series of Γ(−α − 1, βγ),

which requires βγ  1. In the opposite limit, βγ  1, we may still use the low-temperature expansion (6), even if β  1. Singularities occurring at integer α in (7) cancel if ε-expanded.

If γ₁ = 1, β  1 and α > 1, the maximum of p^T,L(ω; γ₁, n₁)_α, β, cf. (5), is determined by the lead factor in the high-temperature expansion (7). This peak is located at and is followed by power-law decay ∝ω^−α for ω  m_t/β and exponential decay starting at about ω ≈  m_t/β. The low-temperature expansion (6) applies for ω  m_t/β. If γ₁  1 (but still βγ₁  1 and α > 1), the peak of p^T,L(ω;γ₁, n₁)_α, β is determined by the factor in (7) and occurs at ω_max ≈ m_t. It is followed by a broken power-law, at first p^T,L_α, β ∝ 1/ω, in the range 1  ω/m_t  γ₁, and then p^T,L_α, β ∝ ω^−α in the interval γ₁  ω/m_t  1/β. At ω ≈ m_t/β, there is an exponential cutoff according to (6).

The spectral fit is performed with the E²-rescaled flux density E²dN^T,L/dE, which relates to the energy density p^T,L_α, β in (5) as

where d is the distance to the pulsar. The preceding scaling relations for p^T,L_α, β give a plateau value E²dN^T,L/dE ∝ 1 in the range 1  E/(m_t c²)  γ₁, followed by power-law decay E²dN^T,L/dE ∝ E^1−α in the band γ₁  E/(m_t c²)  1/β, and subsequent exponential decay.

In the foregoing discussion of the high-temperature asymptotics, we assumed an electron index α > 1. We still have to settle the case α < 1. The spectral peak at ω_max ≈ m_t is again determined by the factor . Adjacent is a power-law slope p^T,L_α,β ∝ 1/ω in the range 1  ω/m_t   1/β, so that E²dN^T,L/dE ∝ 1. At ω ≈  m_t/β, there is the cross-over to exponential decay, cf. (6). This holds true for γ₁ = 1 as well as γ₁  1, provided βγ₁  1, since the leading order of the high-temperature expansion (7) does not depend on γ for α < 1. If βγ₁  1 despite of β being small, then the low-temperature expansion (6) applies instead of (7).

3. Tachyonic cascade spectra and spectral fits

Fig. 1, Fig. 2 and Fig. 3 show the E²-scaled differential number flux, E²dN^T+L/dE, as defined in (8), fitted to the data points. As the polarization is not observed, we add the transversal and longitudinal densities, writing p^T+L  p^T + p^L for the energy density and dN^T+L  dN^T + dN^L for the number flux. We restore the natural units on the right-hand side of (8), up to now we have used  = c = 1, and write E₍₁₎ for ω₍₁₎. The spectral maps are in MeV units, with the differential tachyon flux dN^T,L/dE in units of MeV⁻¹s⁻¹cm⁻². The normalization factor in (5) is dimensionless, , where n₁ is the number of radiating electrons with Lorentz factors exceeding γ₁, distributed according to density (2). α is the electronic power-law index, and β the exponential cutoff in the electron spectrum, both to be determined from the spectral fit like n₁ and γ₁. As for the electron count n₁, it is convenient to use a rescaled parameter for the fit,

which determines the amplitude of the tachyon flux. Here, [keV s] implies the tachyon mass in keV units, that is, we put m_tc² ≈ 2.15 in the spectral function (4). The purpose of this rescaling is to avoid the distance estimate in the actual fitting procedure. Once is extracted from the spectral fit, we can calculate via (9) the number of radiating electrons n₁ constituting the density dρ_α,β. This, however, implies that all tachyons reaching the detector are properly counted. At γ-ray energies, only a tiny α_q/α_e-fraction (the ratio of tachyonic and electric fine structure constant, α_q/α_e ≈ 1.4 × 10⁻¹¹) of the tachyon flux is actually absorbed, cf. [6] and Section 4. This requires a rescaling of the electron count n₁, so that the actual number of radiating electrons is . In Table 1, we list the flux amplitude inferred from the spectral fit, as well as the renormalized electron count . (In the table, we drop the subscript 1 in and .) The density dρ_α,β is defined by parameters obtained from the spectral fit.

In the case of a thermal electron distribution (α = −2), we may identify kT [TeV] ≈ 5.11 × 10⁻⁷/β. We use this definition of electron temperature also for other electron indices α. kT is the energy at which the exponential decay of the electron density (2) sets in. At high temperature, density (2) is peaked at γ_peak ≈ −α/β, provided α < 0 and β  α; the peak is followed by exponential decay. If α > 0, the electron density is decreasing monotonically, at first as power-law and exponentially beyond kT. Nearly straight power-law slopes, as seen in the low GeV range of Fig. 1, can only emerge for electron indices α > 1, cf. after (7). The spectral break from the MeV plateau into the power-law slope is determined by the lower edge γ₁ of electronic Lorentz factors, cf. after (4). If α < 1, the MeV plateau is followed by exponential decay, without power-law cross-over, as exemplified by the thermal cascade (T + L)_B in Fig. 3.

Wideband cascade spectra are obtained by averaging over multiple electron populations,

Similar linear combinations, including pure power-laws, have been used to model γ-ray burst spectra [20] and [21] and electron distributions in galaxy clusters [22]. The normalization factors A_αi,βi(γ_i,n_i) of the individual averages in this series are calculated via

where n_i is the electron count and γ_i the smallest Lorentz factor in the population defined by dρ_αi,βi. The electron numbers n_i (attached to a power-law index α_i, temperature β_i = mc²/(kT_i), and interval boundary γ_i) determine the weights in the linear combination (10).

We summarize the flux density plotted in Fig. 1 and Fig. 3. The spectral fit is done for unpolarized radiation, so that we start by summing over the polarization components,

We plot the unpolarized E²-scaled flux density E²dN/dE, as well as the polarization components E²dN^T,L/dE. (We occasionally use a superscript T + L to indicate the total flux density, writing dN^T+L instead of dN.) The polarization components in (12) are obtained by summing over two cascades, cf. (8) and (10),

The individual cascades used in the least-squares fit described in the caption of Fig. 2 are assembled from (2), (4), (5) and (8). Restoring dimensions, we find their explicit shape as

Each cascade depends on four fitting parameters, α_i, β_i, γ_i, n_i, whose meaning is explained at the beginning of this section. [The actual spectral fit described in the figure captions is done by at first summing over the polarizations in (14) and by adding the χ²-fits of the unpolarized cascades E²dN^T+L(E;α_i, β_i, γ_i, n_i)/dE. That is, we interchange the summations in (12) and (13) to obtain E²dN/dE.] θ is the Heaviside step function, and we use the shortcut

where we put Δ^T = 1 and Δ^L = 0, for transversal and longitudinal polarization, respectively.

The total tachyonic flux density E²dN/dE is plotted in Fig. 1 and Fig. 3 (solid line T + L). The individual unpolarized cascades, E²dN^T+L(E;α_i, β_i, γ_i, n_i)/dE, obtained by summing over the polarizations in (14), are labeled ρ_i=1,2 (dashed curves). They add up to the total tachyonic flux density labeled T + L. The electron densities ρ_i in Table 1 produce the cascade spectra ρ_i and are in turn defined by parameters obtained from the least-squares fit.

The cutoff kT in the electron energy is listed in Table 1, for each electron population generating a cascade in the γ-ray wideband. In the case of protonic source particles, the cutoff energies have to be multiplied with 1.8 × 10³, the proton/electron mass ratio. These cutoffs are to be compared to two spectral breaks in the cosmic-ray spectrum, occurring at 10^3.5 TeV and 10^5.8 TeV, dubbed knee and second knee, respectively [23]. A protonic source density generating the ρ₁-cascade has a cutoff at about 10^4.6 TeV, suggesting protons in the pulsar magnetosphere with energies between the first and second knee.

4. Conclusion

We have considered superluminal spectral densities generated by freely moving electrons, discussed the spectral averaging with electronic source distributions, and derived the high- and low-temperature expansions of the averaged radiation densities. We demonstrated that the γ-ray flux of the Crab pulsar can be reproduced by tachyonic cascade spectra, and explained the spectral plateau, the power-law slopes, and the spectral curvature.

Here, we have studied superluminal radiation from electrons in uniform motion. Tachyonic synchrotron and cyclotron radiation was investigated in [24] and [25]. In the zero magnetic field limit, the tachyonic synchrotron densities converge to the spectral densities (1) for uniform motion. Electromagnetic synchrotron radiation vanishes in this limit, of course. A finite bending radius induces modulations in the spectral densities. These oscillations are quite small, just tiny ripples along the spectral slope with increasing amplitude toward the spectral break [25]. If integrated over a thermal or power-law electron distribution as done here, these oscillations are averaged out.

Several electromagnetic radiation mechanisms have been invoked to model the spectra of γ-ray pulsars, most notably curvature radiation due to electric fields followed by pair creation, accompanied by synchrotron radiation and inverse Compton scattering [26]. These mechanisms are quite interlocked, cf. the flow-chart in [27], and subject to frequent revisions [28] and [29]. An electromagnetic spectral fit to the pulsed Crab emission, directly comparable to Fig. 1, can be found in Fig. 11 of [28].

In Section 3, we pointed out that only a fraction α_q/α_e ≈ 1.4 × 10⁻¹¹ of the tachyon flux arriving at the detector is actually absorbed, so that the measured flux has to be rescaled with the ratio of electric and tachyonic fine structure constant. This rescaling applies to γ-ray spectra only, to frequencies much higher than the 2 keV tachyon mass, so that the mass-square can be dropped in the tachyonic dispersion relation. At X-ray energies, we have to rescale with the respective cross-section ratio, σ_e(ω)/σ_q(ω), where σ_e is the electromagnetic cross-section and σ_q its tachyonic counterpart. The detection mechanism for hard and high-energy X-rays is usually ionization, generating scintillations in NaI(Tl) and CsI(Na) crystals of phoswich detectors [30], or gas scintillations by ionization of Xenon atoms in proportional counters [31]. Tachyonic ionization cross-sections of hydrogen-like ground states, that can be used for X- as well as γ-rays, have been calculated in [32]. At γ-ray energies, photonic and tachyonic dispersion relation coincide, and the cross-section ratio of the detection mechanism (such as ionization [33], Compton scattering [34] and pair production [35], the latter two quantified by Klein-Nishina and Bethe–Heitler cross-section) approaches a constant limit value α_e/α_q, which amounts to an overall rescaling of the tachyonic flux amplitude. By contrast, at X-ray energies, we would have to rescale the tachyonic flux density with the energy dependent cross-section ratio, which affects the shape of the spectral map. If a grating spectrometer is used, the interference peaks are determined by wavelength. In this case, the observational plots have to be reparametrized by wavelength, and then the tachyonic dispersion relation is substituted to obtain the actual energy-flux relation.

Acknowledgement

The author acknowledges the support of the Japan Society for the Promotion of Science. The hospitality and stimulating atmosphere of the Centre for Nonlinear Dynamics, Bharathidasan University, Trichy, and the Institute of Mathematical Sciences, Chennai, are likewise gratefully acknowledged.